Support montgomery form directly in non-native field operations

[huitseeker]

Many implementations of elliptic curves in libraries (e.g. bls12_381) internally make use of the montgomery form to represent field elements, where a number a mod P is stored as aR mod P for some factor R (for bls12_381, this is 2^384 mod P).

Currently, the existing non-native field operations supported all simply deal with regular a mod P representations of these numbers. In the case of addition and subtraction nothing changes, but when multiplication and division are involved, the R factor needs to be dealt with separately. This incurs overhead by requiring multiple additional operations. It would be desirable to have non-native field operations that natively directly support the montgomery form when performing multiplication and division/inversion.

This is not a high-priority blocking issue since we can work around this issue by using a field multiplication precompile to perform the reduction, i.e. multiplying by R^(-1) to remove the R factor, perform the usual operation, then multiply back by R to return it to its montgomery representation. There is still overhead in doing this though, specially in long-running computations that perform many of these roundtrips.